Aggregate labor market outcomes: The roles of choice and chance

Transcription

1 Quantitative Economics 1(2010), / Aggregate labor market outcomes: The roles of choice and chance PER KRUSELL IIES, CAERP, CEPR, and NBER TOSHIHIKO MUKOYAMA University of Virginia and CIREQ RICHARD ROGERSON Arizona State University and NBER AYŞEGÜL ŞAHIN Federal Reserve Bank of New York Commonly used frictional models of the labor market imply that changes in frictions have large effects on steady state employment and unemployment. We use a model that features both frictions and an operative labor supply margin to examine the robustness of this feature to the inclusion of an empirically reasonable labor supply channel. The response of unemployment to changes in frictions is similar in both models, but the labor supply response present in our model greatly attenuates the effects of frictions on steady state employment relative to the simplest matching model and two common extensions. We also find that the presence of empirically plausible frictions has virtually no impact on the response of aggregate employment to taxes. KEYWORDS.Laborsupply,labormarketfrictions,taxes. JEL CLASSIFICATION.E24,J22,J64. Per Krusell: Per.Krusell@iies.su.se Toshihiko Mukoyama: tm5hs@virginia.edu Richard Rogerson: Richard.Rogerson@asu.edu Ayşegül Şahin: Aysegul.Sahin@ny.frb.org Some of the material in this paper was previously contained in our paper titled Labor supply in a frictional labor market. We thank the editor, Nicola Pavoni, Naoki Shintoyo, and two anonymous referees for useful comments, as well as seminar participants at CEMFI, Columbia, Universitat Autonoma de Barcelona, USC, UCLA, the Atlanta Fed, Yale, UC Riverside, Queen s, Georgetown, Wharton, Toronto, Ryerson, Western Ontario, Stockholm School of Economics, IIES, University of Oslo, Concordia, Northern Illinois, Yonsei University, Korea University, Ohio State, Notre Dame, Penn State, Indiana, Maryland, and George Washington, as well as conference participants at the Labor Markets Conference at the IFS (2008), the Winter Institute of Macroeconomics 2009 (Tokyo), the Bewley Conference at UT Austin, and the 2009 SED Conference. We thank Joe Song for excellent research assistance. Krusell thanks the NSF for financial support, Mukoyama thanks the Bankard Fund for Political Economy for financial support, and Rogerson thanks the NSF and the Korean Science Foundation (WCU-R ) for financial support. The views expressed in this article are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of New York or the Federal Reserve System. Copyright 2010 Per Krusell, Toshihiko Mukoyama, Richard Rogerson, and Ayşegül Şahin. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at DOI: /QE7

2 98 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) 1. INTRODUCTION Two frameworks dominate analyses of aggregate employment frictionless models that follow in the tradition of Kydland and Prescott (1982), and frictional models in the tradition of Mortensen and Pissarides (1994). While frictionless modelsnecessarily imply that changes in employment entirely reflect changes in desired labor supply (i.e., choice), the simplest specifications of frictional models imply that changes in employment entirely reflect changes in the probability of receiving an offer (i.e., chance). It seems clear that both choice and chance influence individual employment outcomes in reality; that is, at any point in time some individuals are not employed by choice, while others are not employed by chance. There is good reason to believe that these two features have interesting interactions, in that the presence of frictions may attenuate the labor supply responses in frictionless models, while the presence of an operative labor supply margin might similarly attenuate the effects of frictions. In this paper, we assess the relative importance of these two forces in shaping aggregate steady state employment. We carry out this analysis in the model of Krusell, Mukoyama, Rogerson, and Şahin (2009). That paper built an empirically reasonable model that includes frictions and an operative labor supply margin that exhibits plausible income and substitution effects. In considering the implications of the model for labor market flows, we adopt a more general view of unemployment than adopted by official statistical agencies. Specifically, we assign an individual to this state, which we refer to as generalized unemployment, if they are not working but answer yes to the question of whether they would like to work at the market wage rate. This concept of unemployment arises naturally in our model and can be connected to the data gathered by the Current Population Survey. Our emphasis on the desire to work as opposed to active search for work is motivated both by the work of Jones and Riddell (1999, 2006) regardingmarginallyattachedworkers the fact that active search seems to imply relatively little time use and the fact that many workers find jobs through friends and relatives. We use this model to ask two simple questions about the forces that influence steady state employment and unemployment. The first question is how changes in frictions affect aggregate steady state outcomes. In the simplest matching model, the level of frictions (captured by both the offer arrival rate and the separation rate) critically affects the level of both aggregate employment and unemployment. We assess the extent to which this is altered when one embeds the frictional model in a context where individuals also solve an empirically reasonable lifetime labor supply problem. Intuitively, if employment opportunities are harder to come by (or jobs do not last very long), individuals can adjust lifetime labor supply by extending the length of employment spells when employed or by accepting more employment opportunities when not employed. In our calibrated model, we find that the increase in unemployment is very similar to that implied by a simple matching model. In contrast, we find that labor supply responses greatly attenuate the direct effect of frictions on employment. Specifically,we find that a decrease in the arrival rate of employment opportunities leads to a large increase in the unemployment rate, but only a small decrease in the employment rate. We conclude that while the role of frictions for steady state aggregate unemployment

3 Quantitative Economics 1 (2010) Aggregate labor market outcomes 99 seems robust to adding a nondegenerate labor supply decision, the impact of frictions on steady state employment is significantly overstated in simple matching models. We show that this same result holds in versions of the matching model in which match formation and match termination decisions are introduced. While these extensions add an element of choice to the simplest matching model, we show that incorporating empirically reasonable income and substitution effects is of first order importance in assessing the quantitative effects of frictions on employment. Second, we use our model to assess the effects of increases in labor taxes used to fund lump-sum transfers. This issue, recently examined in a frictionless model by Prescott (2004), seems to be a simple and sharp example of how an operative labor supply margin influences steady state employment. We ask how these effects are altered by the presence of reasonable frictions. We find that the employment effects are effectively unchanged by the presence of frictions. This result holds not only for tax increases, but also tax decreases, which is perhaps more interesting since it is more likely that frictions will interfere with the desire to increase the fraction of time spent in employment. 1 We conclude that frictions do not seem to be of first order importance in the determination of steady state employment. Interestingly, in our model with frictions, higher taxes lead to both higher unemployment and higher nonparticipation, even holding the level of frictions constant. So, although the aggregate effects on employment in our model are effectively those found in the frictionless version of the model, the analysis shows that there are also effects on both the level and nature of unemployment. Our paper is related to many papers in the literature. Merz (1995) andandolfatto (1998) werethefirsttointroducefrictionsintoanotherwisestandardversionofthe growth model. A key feature of these models is that employment is completely determined by frictions, just as in simple frictional models. The model in Alvarez and Veracierto (1999)isclosertoours,sinceitfeaturesbothastandardlabor leisurechoiceand frictions, but it cannot be calibrated to match worker flows since worker flows are indeterminate in their equilibrium. Moreover, they do not ask the question of how changes in frictions affect steady state outcomes. Garibaldi and Wasmer (2005) developedathreestate model of the labor market that matches flows between the three states, but their model assumes linear utility. Ljungqvist and Sargent (2006, 2008) considered models that feature indivisible labor and frictions, but do not consider the impact of frictions on aggregate employment. Low, Meghir, and Pistaferri (forthcoming) considered a model with frictions and a nondegenerate labor supply decision. They considered a richer model of frictions and income support programs, but their analysis is partial equilibrium and they address very different issues than we do. An outline of the paper follows. Section 2 describes the model. Section 3 describes the calibration of the model and presents the implications of the calibrated model for labor market flows. Section 4 analyzes the effects of changes in frictions on steady state outcomes in a simple matching model, while Section 5 considers the same issue in some extensions of this model. Section 6 presents the results for analyzing tax and transfer programs. Section 7 discusses robustness and Section 8 discusses some issues regarding the nature of idiosyncratic shocks. Section 9 concludes. 1 AsimilarresultwasfoundinKrusell,Mukoyama,Rogerson,andŞahin (2008), but the calibrated model in that paper was not consistent with worker flows.

4 100 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) 2. MODEL The economy is basically that in Krusell et al. (2009). 2 The model has two key features. First, abstracting from frictions, it features a canonical dynamic model of labor supply in a context that features idiosyncratic shocks and incomplete markets for both credit and risk-sharing. 3 As documented in Krusell et al. (2009), idiosyncratic shocks are an essential element in allowing the model to match the key features of labor market flows. While we could carry out our analysis in the context of a model that assumed complete markets, we believe that the incomplete markets model has become a more useful benchmark for quantitative work in models that feature idiosyncratic shocks. The frictions that we focus on are the two key frictions that characterize steady state labor market outcomes in almost any search and matching model: a job separation rate for employed individuals and an offer arrival rate for nonemployed individuals. While our model is somewhat minimalist, it seems a natural benchmark for an initial quantitative assessment of how frictions interact with labor supply. The economy is populated by a continuum of workers with total mass equal to 1. All workers have identical preferences over streams of consumption and time devoted to work given by β t [log(c t ) αe t ] t=0 where c t 0 is consumption in period t, e t {0 1} is time devoted to work in period t, 0 <β<1 is the discount factor, and α>0 is the disutility of work. Our preference specification assumes offsetting income and substitution effects, which we think represents a natural benchmark. 4 Individuals are subject to idiosyncratic shocks that affect the static payoffs of working relative to not working. While many shocks may have this property, for example, shocks to market opportunities, shocks to home production opportunities, health shocks, family shocks, preference shocks,..., we represent the net effect of all of these shocks as a single shock and model it as a shock to the return to market work. In particular, letting s t denote the quantity of labor services that people contribute if working, we assume an AR(1) stochastic process in logs, log s t+1 = ρ log s t + ε t+1 2 The model is essentially that of Krusell et al. (2009). One difference is that here we assume that a separated worker must spend at least one period out of employment. The earlier paper does not perform any analysis of how frictions or taxes affect steady state outcomes. We also explore some alternative calibrations here. 3 Several papers have recently analyzed labor supply in models with idiosyncratic shocks and incomplete markets, including Floden and Linde (2001), Low (2005), Domeij and Floden (2006), Pijoan-Mas (2006), and Chang and Kim (2006, 2007). Relative to these papers, the distinguishing feature of our model is the presence of frictions. 4 The search and matching literature commonly assumes linear utility. Relative to the linear specification, the key feature of our specification is a much more substantial income effect. To the extent that the balanced growth specification is challenged by some microstudies, the issue is that the income effect is perhaps even larger (see, for example, Hall (2009)).

5 Quantitative Economics 1 (2010) Aggregate labor market outcomes 101 where the innovation ε t is a mean zero normally distributed random variable with standard deviation σ ε. This process is the same for all workers, but realizations are independent and identically distributed (i.i.d.) across workers. We formulate equilibrium recursively and focus solely on the steady state equilibrium. In each period there are markets for output, capital services, and labor services, but there are no insurance markets, so individuals will (potentially) accumulate assets to self-insure. We normalize the price of output to equal 1 in all periods, and let r and w denote steady state rental rates for a unit of capital and a unit of labor services, respectively. If a worker with productivity s choosesto work, then he or she would contribute s units of labor services and therefore earn ws in labor income. We assume that individual capital holdings must be nonnegative or, equivalently, that individuals are not allowed to borrow. There is a government that taxes labor income at constant rate τ and uses the proceeds to finance a lump-sum transfer payment T subject to a period-by-period balanced budget constraint. In a later section, we also consider a stylized unemployment insurance system. In steady state, the period budget equation for an individual with k t units of capital and productivity s t is given by c t + k t+1 = rk t + (1 τ)ws t e t + (1 δ)k t + T The production technology is described by a Cobb Douglas aggregate production function Y t = Kt θ L1 θ t where K t is aggregate input of capital services and L t is aggregate input of labor services: K t = k it di L t = e it s it di Output can be used as either consumption or investment, and capital depreciates at rate δ. Frictions in the labor market are captured by two parameters, λ w and σ, whereλ w is the employment opportunity arrival rate and σ is the employment separation rate. Specifically, we assume there are two islands which we label as the production island and the leisure island. At the end of period t 1,anindividualiseitherontheproduction island or on the leisure island, depending on whether he or she worked during the period. At the beginning of period t, each individual will observe the realizations of several shocks. First, each worker receives a new realization for the value of his or her idiosyncratic productivity shock. Second, each individual on the production island observes the realization of an i.i.d. separation shock: with probability σ, the individual is relocated to the leisure island. At the same time, each individual who began the period on the leisure island observes the realization of an i.i.d. employment opportunity shock: with probability λ w, an individual is relocated to the production island. In terms of connecting our model with the literature, it is intuitive to think of σ as the exogenous job separation

6 102 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) rate and λ w as the exogenous job arrival rate. Note that given our timing assumption, an individual who suffers the σ shock will necessarily spend at least one period in nonemployment. Once the shocks have been realized, individuals make their labor supply and consumption decisions, although only workers with an employment opportunity can choose e = 1.Anindividualontheproductionislandwhochoosesnottoworkwillthen be relocated to the leisure island at the end of period t and will therefore not have the opportunity to return to the production island until receiving a favorable employment opportunity shock. An individual s state consists of his or her location at the time that the labor supply decision needs to be made, the level of asset holdings, and productivity. Let W(k s)be the maximum value for an individual who works and let N(k s) be the maximum value for an individual who does not work given that he or she has productivity s and capital holdings k.define V(k s)= max{w(k s) N(k s)} The Bellman equations for W and N are given by W(k s)= max c k { log(c) α + βes [(1 σ)v (k s ) + σn(k s )] } s.t. c + k = rk + (1 τ)ws + (1 δ)k + T c 0 k 0 and N(k s) = max c k { log(c) + βes [λ w V(k s ) + (1 λ w )N(k s )] } s.t. c + k = rk + (1 δ)k + T c 0 k 0 3. CALIBRATION We calibrate the model as in Krusell et al. (2009), so we refer the reader to that paper for a more detailed analysis and discussion of the calibration. A key aspect of the calibration procedure is to choose parameters so that the distribution of workers across states and the flows of workers between states are similar to those in the U.S. economy. Official statistics divide nonemployed workers into the two categories of unemployed and out of the labor force based primarily on how they answer a question regarding active search in the previous 4 weeks. Krusell et al. (2009) arguedthatadifferentcriterionis more natural, and we use their criterion to assign the nonemployed into two mutually exclusive groups. This assignment is based on how an individual responds to the question of whether they would like to work if work were available. The group who answers yes to this question is larger than the group who is unemployed based on the standard definition. We will refer to our categories as representing generalized notions of unemployment and nonparticipation: to remind the reader of this difference we will use E to denote the employment state, U G to denote the generalized unemployment state, and N G to denote the generalized not in the labor force state. To have a consistent definition in the model and the data, we use this same criterion to define generalized unemployment in the data. The generalized unemployment rate is somewhat larger than

7 Quantitative Economics 1 (2010) Aggregate labor market outcomes 103 the official unemployment rate (8 4% versus 5 1%). We also compute flows among labor market states based on our definitions of the three states. As a practical matter, it turns out that this adjustment has very little effect on the transition probabilities. An additional issue to address in calibrating the model concerns the population that the model best captures. Our model has single agent households that live forever. In reality, people have finite lives and often live in multimember households. In Krusell et al. (2009), we presented transition probabilities for different choices of the underlying population sample, and argued that the model s ability to account for the flows is not particularly affected by the choice of the underlying population, although the fit is somewhat better for males than females. We present results for calibrations based on two different population samples. For our benchmark calibration, we take a broad interpretation of the model and use it to capture labor market flows of all individuals older than 16. In Section 7, weconsideranarrowerinterpretation,inwhichwecalibratethemodelto flows for males aged 25 54, and show that our key conclusions hold for this alternative calibration. Having described how we will measure flows across states in the data and the model, we now consider how to calibrate the model s parameters. The model has nine parameters that need to be assigned: preference parameters β and α,productionparametersθ and δ, idiosyncratic shock parameters ρ and σ ε,frictionalparametersσ and λ w,and the tax rate τ. Thelengthofaperiodissetto1month.Becauseourmodelisavariation of the standard growth model, we can choose some of these parameter values using the same procedure that is typically used to calibrate versions of the growth model. The features of incomplete markets and uncertainty imply that we cannot derive analytic expressions for the steady state, and so cannot isolate the connection between certain parameters and target values. Nonetheless, it is still useful and intuitive to associate particular targets and parameter values. Specifically, given values for λ w, σ, ρ, andσ ε,we choose θ = 3 to target a capital share of 3, δ to achieve an investment to output ratio equal to 2,anddiscountfactorβ to target an annual real rate of return on capital equal to 4%. Theotherpreferenceparameterα, whichcapturesthedisutilityofworking,is set so that the steady state value of employment is equal to 633. Thisisthevalueof the employment to population ratio for the population aged 16 and older for the period The tax rate is set at τ = 30. Following the work of Mendoza, Razin, and Tesar (1994), there are several papers which produce estimates of the average effective tax rate on labor income across countries. Examples include Prescott (2004) and McDaniel (2006). There are minor variations in methods across these studies, which do produce some small differences in the estimates, and the value.30 is chosen as representative of these estimates. 6 5 We calibrate to values for the period because this is the period for which we have consistent measures of labor market flows. 6 Note that Prescott (2004) made an adjustment to the average labor tax rate to arrive at a marginal tax rate that is roughly 40%.Forthepurposeofcomputingtheeffectofchangesintaxes,thisadjustmentplays no role.

8 104 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) It remains to choose values for λ w, σ, ρ, andσ ε.wechooseλ w so that the steady state generalized unemployment rate in our model (i.e., U G /(E + U G ))isequalto 084, which is the average value for the generalized unemployment rate in the U.S. data for the period We choose σ to target the flow rate from employment to generalized unemployment. Krusell et al. (2009)showedthattheabilityofthemodeltoaccountfortheflowsbetween states remains relatively constant for a wide range of values of ρ and σ ε.what mattered most was that ρ was reasonably persistent (at least 5), but not too close to being a unit root (say less than 97), and that σ ε was not too small. In their benchmark calibration, they assumed ρ = 92 and σ ε = 21 expressed on an annual basis. These values correspond to one set of estimates of idiosyncratic wage shocks for prime-aged working males, asreported inflodenand Linde (2001). A key issue for our quantitative exercises is the extent to which different specifications of the shock process influence our results, despite having little impact on worker flows. It turns out that the results are relatively unaffected by considering different calibrated values for ρ and σ ε,giventhatineach case we recalibrate the remaining parameters to continue to hit the same targets. As a result, we will only present results for this one set of values for ρ and σ ε.table1 shows our calibrated parameter values. The labor market flows in our calibrated model and the data are displayed in Table 2. A major discrepancy has to do with the flow of workers from U G to N G.Asdiscussed in Krusell et al. (2009), this discrepancy is much less if we consider male workers aged or instead of the whole population, as we will see in our alternative calibration presented later. Additionally, assuming some survey response error that causes spurious transitions between N G and U G also removes much of the discrepancy. 7 While there is room for additional improvements relative to this simple model, we feel that the match is sufficiently close to justify using this model to revisit some basic questions about the forces that shape steady state employment and unemployment. Even accounting for survey response error as noted above, the flow rate from U G to E is somewhat high relative to the data. If one is concerned about the calibrated level of frictions being reasonable, this might be viewed as an important target. Krusell et al. (2009) alsopresentedanalternativecalibrationinwhichtheflowratefromu G to E is TABLE 1. Benchmark calibration. Targets I Y = 20, rk Y = 3, E P = 633, U G E+U G = 084, r δ = 04, E U G = 021 Parameter Values θ δ β α ρ σ ε λ w σ τ Survey response error also lowers the measured flow rate from U G to E since some of the people counted in U G are actually in N G and therefore transition to E with much lower probability.

9 Quantitative Economics 1 (2010) Aggregate labor market outcomes 105 Adjusted U.S TABLE 2. Flows in the model and data. To Model From E U G N G From E U G N G E E U G U G N G N G To targeted instead of the stock of U G. While we do not report any results for this alternative calibration, we note here that all of the results presented below are effectively identical for this alternative calibration. 8 Although our calibration targets flows for the entire population, we can still contrast outcomes across different subgroups of the population in our model. Of particular relevance is the distinction between individuals who vary in their degree of attachment to the labor force. One simple measure of this in the model is the individual s value of the idiosyncratic shock s. Inafrictionlessversionofourmodelwithcompletemarkets, individuals would work when s is above some threshold. If we look at different parts of the productivity distribution, we see very different behavior of individuals. In what follows, we will contrast how behavior changes in the upper part of the productivity distribution with that of the overall population. Specifically, we will focus on the highest 42% of the distribution. In steady state, the breakdown of this group between the three states E, U G,andN G,is 920, 056,and 024.Thekeyfeatureofthisgroupisthatvirtually everyone in this group wants to work. We will see later on that this group exhibits very different responses than the aggregate. Because wealth accumulation is a key element in shaping labor supply responses of individuals in our model, it is important to assess the extent to which the wealth distribution in our model is similar to that in the data. Table 3 shows the share of wealth held by various groups in the population, both for our model and for the United States, as well as the Gini coefficient. The values for the United States are taken from Budria Rodriguez, Diaz-Gimenez, Quadrini, and Rios-Rull (2002). TABLE 3. Share of wealth owned by various groups. Gini Top 1% Top 5% Top 10% Top 20% Top 40% Top 60% Top 80% Model Data Alternatively, if we were to use a slightly smaller value of σ, our calibration procedure would produce a much smaller value for the transition rate from U G to E,sincethestockofworkersinE is almost an order of magnitude larger than the stock in U G.

10 106 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) The basic message from Table 3 is similar to that found in previous analyses of this type. 9 In particular, the model does a reasonable job of capturing the wealth distribution except for the concentration in the upper tail. Note that the presence of our transfer program improves the model s ability to fit the lower part of the wealth distribution relative to models without this feature. The Gini coefficient in the model is about three-quarters as large as it is in the data. Although our model does not capture the existence of individuals like Bill Gates, given the very small size of this group, we believe this failure is not particularly significant from the perspective of understanding aggregate labor supply. 4. FRICTIONS AND THE STEADY STATE: I.ABENCHMARK COMPARISON One of the defining features of the Pissarides matching model and its many variants is that the level of frictions plays a key role in determining not only the level of aggregate unemployment, but also in determining the level of aggregate employment. 10 Intuitively, labor supply considerations will attenuate the impact of changes in frictions on aggregate employment. The reason for this is that if it becomes harder to find employment opportunities, then workers will be more willing to continue with a job opportunity once they find it or decide to accept employment at lower productivities. The goal of this section is to explore the quantitative importance of these effects in our model relative to standard frictional models. We begin by exploring the impact of exogenous changes in the level of λ w,thatis,we evaluate the impact on the steady state of an exogenous change in the level of frictions. We are primarily interested in the extent to which the responses in our model are different than those that would emerge in a benchmark version of the Pissarides matching model. In the simplest Pissarides model, the match separation rate is exogenous, but the job offer arrival rate is endogenously determined by the volume of vacancy posting. In this model, all job offers are accepted, so the job offer arrival rate is also the probability that an unemployed worker becomes employed. If the match separation rate is σ and the job offer arrival rate is λ w,andweassumethataseparatedindividualhastowaitone period before job offers start arriving, then the law of motion for the unemployment rate is u t+1 = (1 λ w )u t + σ(1 u t ) It follows that the steady state employment rate is given by ū = σ λ w + σ We set σ = 022 as in our benchmark calibration and then set λ w so that the steady state unemployment rate is equal to 084, whichwasthesametargetthatwematched 9 See, for example, Krusell and Smith (1998), Budria Rodriguez et al. (2002), and Castaneda, Diaz-Jimenez, and Rios-Rull (2003). 10 See Pissarides (2000) for a variety of models that have this property. In addition, see Krusell, Mukoyama, and Şahin (forthcoming) andthereferencesthereinformodelswhichfeaturecurvature inpref- erences and asset accumulation.

11 Quantitative Economics 1 (2010) Aggregate labor market outcomes 107 TABLE 4. Effect of λ w on employment and generalized unemployment rates. Our Model Pissarides Model E/P U G /(U G + E) E/P U G /(U G + E) λ w = % 5 6% λ w = % 6 1% λ w = % 8 4% λ w = % 8 4% λ w = % 16 0% λ w = % 16 4% in our calibration. The implied value of λ w is 24. We will then consider equal proportional changes in the value of λ w in the two models, that is, we increase or decrease λ w by the same percentage in the two models. Although the value of λ w is endogenously determined in the Pissarides model, we do not model the source of this change. Rather, we focus simply on the consequences of such a change for employment and generalized unemployment. Table 4 shows the effects for the aggregate employment to population ratio (E/P) and the unemployment rate in the two models, for our benchmark calibration. We emphasize that the predictions of our model are very similar for different values of ρ and σ ε,andforthealternativecalibrationprocedureinwhichλ w is targeted to match the E to U G flow. In the interest of space, we only report results for the benchmark calibration, as shown in Table 4. In reading Table 4, eachrowrepresentsthesamepercentagechangeinλ w relative to the two benchmark calibrations, which by construction each have the same unemployment rate. A striking result emerges. If one looks at the responses on generalized unemployment rates, one observes that the effects are very similar across the two different models, especially for the case of decreases in λ w.moreover,theeffectsarelarge: when λ w is decreased from the benchmark setting to the lowest value in the table, the generalized unemployment rate roughly doubles in both cases. But when one looks at the employment to population ratio responses, one sees dramatic differences. In the Pissarides model, changes in the generalized unemployment rate and changes in the employment to population ratio are necessarily mirror images of each other since, by construction, all workers are in the labor force. Hence, the Pissarides model also predicts large employment responses as a result of changes in λ w.insharpcontrast,our model predicts very small changes in employment to population ratios. The change in the employment to population ratio in our model is only about one-seventh as large as the change in the Pissarides model. For example, when moving from the benchmark specification to the lowest value of λ w in the table, the employment to population ratio decreases by 8 percentage points in the Pissarides model, but only slightly more than 1 percentage point in our model. To see why the two models give such different employment responses, it is instructive to examine the durations of employment and generalized unemployment spells, shown in Table 5. Inbothmodels,adecreaseinλ w leads to an increase in the duration of generalized unemployment. The proportional changes are very similar in the two models, but the changes in employment durations are very different. In the Pissarides

12 108 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) Our Model TABLE 5. Effect of λ w on spell durations. Pissarides Model E U G E U G λ w = λ w = λ w = λ w = λ w = λ w = model, a decrease in λ w has no effect on the duration of employment spells. In contrast, in our model, the duration of employment spells increases significantly in response to decreases in λ w. For example, in moving from the benchmark value of λ w to the lowest value in the table, the duration of employment in our model increases by more than one-quarter. Another way to present this information is to see how the transition matrix across the three states is affected. Table 6 shows the transition matrices for the original calibration (λ w = 428)andthelowervalueof 2. Most of the changes are relatively small, with the exception of the flow from U G to E and U G to U G.Consistentwiththepreviouslydescribedpatterns,thereisslightdecrease in the flow from E to N G with a corresponding increase in the transition rate from E to E. It is also useful to see how frictions affect the decision rules in the benchmark model. In steady state, each productivity level is associated with a threshold level for assets, such that the worker desires to works if assets are below this level and does not desire to work if assets are above this level. As productivity increases, so does the threshold value. Figure 1 shows the contour of reservation values and how it shifts as frictions change. As frictions become larger, that is, as λ w decreases, the asset threshold becomes smaller for any given level of productivity. Finally, it is instructive to examine how the distribution of employment across productivity states is influenced by changes in λ w. Figure 2 plots employment to population ratio at each productivity level for various values of λ w.asfrictionsincrease,someemploymentisshiftedfromtherighttailtotheleft tail. Intuitively, if there are no frictions, then all workers with sufficiently high productivity will work, but in the presence of frictions, some of these workers are not able to work because they do not have an employment opportunity. It is interesting to note that even for a very large change in frictions, the increase in mass at the bottom of the productivity TABLE 6. Transition rates for different levels of frictions. λ w = 428 To λ w = 20 From E U G N G From E U G N G E E U G U G N G N G To

13 Quantitative Economics 1 (2010) Aggregate labor market outcomes 109 FIGURE 1. Effect of frictions on the employment decision rule. distribution is quite small, and it remains true that the lowest productivity workers do not work at all. 11 It is also of interest to contrast outcomes for the sample of high productivity individuals that we described earlier with those for the aggregates shown in Table 4.Table7 shows the results for this group. The interesting result that emerges is that this group responds very much like the aggregates from the Pissarides model. In particular, for this group, the responses in E and U G are close to mirror images of each other. This suggests that while the Pissarides FIGURE 2. Frictions and employment to population ratios. 11 It is important to keep in mind that our model includes a government transfer program, so that individuals do receive some income even when not working.

14 110 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) TABLE 7. Effect of λ w. High Productivity E/P U/(U + E) λ w = % 3 9% λ w = % 5 6% λ w = % 11 3% model is an appropriate model for describing what happens to steady state outcomes for a particular subgroup of the population, it is not appropriate for understanding changes at a more aggregate level. We can also repeat the above analysis to examine how the two different models respond to exogenous changes in σ, the separation shock.proceeding as above,table 8 presents the effects on employment and generalized unemployment. 12 The results are very similar to those found for changes in λ w.changesingeneralizedunemployment rates in response to changes in σ are about one half as large in our model as in the Pissarides model and the employment response is again only about one-seventh as large in our model as in the Pissarides model. Table 9 shows that the reason for the large differences in employment rate responses has to do with a labor supply effect that produces offsetting changes in employment durations. In the Pissarides model, changes in σ lead mechanically to changes in employment duration. The implication is that the large changes in σ are associated with large and proportional changes in employment duration. In contrast, in our model, decreases in σ lead to what, in comparison, are only TABLE 8. Effect of σ on employment and generalized unemployment rates. Our Model Pissarides Model E/P U G /(U G + E) E/P U G /(U G + E) σ = % 6 2% 96 0% 4 0% σ = % 8 4% 91 6% 8 4% σ = % 9 9% 88 9% 11 1% TABLE 9. Effect of σ on spell durations. Our Model Pissarides Model E U G E U G σ = σ = σ = Because the values of σ are the same in the two benchmark economies, we now consider equal changes in the two economies.

15 Quantitative Economics 1 (2010) Aggregate labor market outcomes 111 very moderate increases in employment duration. This is due to a labor supply response. When σ is high, it is less likely that an individual has an employment opportunity in any given period, and as a result they respond by being willing to work at lower productivity levels. This in turn implies less voluntary separations. But when σ decreases, the reverse is true, and individuals become choosier about when to work, leading to more voluntary separations. To close this section we think it is important to emphasize the important role of asset accumulation in generating the above results. In particular, adding curvature to the utility function is not sufficient for capturing the effects that we are stressing. To see this, note that without asset accumulation, the individual s desire to work would depend purely on the level of the idiosyncratic shock and, in particular, not on the level of frictions. Frictions would merely influence the probability of working conditional on wanting to work. Because of this, there is no labor supply response due to a change in frictions, and the key results we stress would disappear. 5. FRICTIONS AND THE STEADY STATE: II.EXTENSIONS One of the key findings in the previous section was that labor supply responses greatly attenuate the effect of frictions on steady state employment. Readers familiar with the matching literature might reasonably argue that this result is heavily influenced by our choice of the benchmark model. In particular, a key feature of the benchmark model is that given the level of frictions, there are no other margins of adjustment that work to at least partially offset the effect of frictions. While this is a property of the benchmark model, simple and popular extensions of this model, including those in Pissarides (1985) and Mortensen and Pissarides (1994), doincludeanadditional choicemarginthatmight play the same role as the labor supply channel in our model. In Pissarides (1985), when a meeting between a worker and a firm occurs, the pair receives a random draw of a permanent match quality. The optimal decision about whether to form a match is characterized by a reservation rule, that is, proceed with forming the match if the draw for match quality is above some threshold. In this setting, steady state employment will depend on both the level of frictions and the reservation value. Intuitively, if frictions become more severe, workers will become less choosy about which matches to form, thereby giving rise to a force that can at least partially offset the direct effect of more severe frictions. Similarly, in Mortensen and Pissarides, all matches start at the same high level for match quality, but this quality subsequently evolves stochastically. In this setting a key decision is when to terminate a match, which under fairly standard conditions will again be characterized by a reservation rule. In this setting an increase in frictions can lead to alowerreservationvalue,implyingthatmatchterminationswilldecrease.onceagain, this creates an opposing effect on steady state employment relative to the direct effect. The question that we ask in this section is whether our finding about the quantitative importance of the labor supply channel for employment responses remains when we compare our model with either of these extensions. We note at the outset that although one naturally expects these extensions to lessen the difference between our model and

16 112 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) the benchmark Pissarides model, there is good reason to think that a significant difference will remain. This is due to the specification of linear utility in the matching literature. In a standard model of labor supply, including the model we are using in this paper, when a worker decreases the fraction of life spent in employment, average consumption decreases, which in turn increases the marginal utility of consumption and creates an incentive for the individual to increase the fraction of time spent in employment. In contrast, a decrease in average consumption in a model with linear utility does not increase the marginal utility of consumption and, therefore, does not contain this force leading to higher employment. In fact, we will find that our labor supply channel is a substantially more powerful offsetting force than the margins present in either of these extensions. We conclude that to assess the quantitative effects of frictions on steady state employment, it is important to not only have the presence of labor supply margins, but also that the labor supply decision exhibit reasonable income and substitution effects. We proceed to describe in more detail each of the two extensions described above. 5.1 Extension 1: Adding a match formation decision In this subsection we analyze an extension to incorporate match quality. In the spirit of our earlier calculation, we consider a continuum of workers, each of whom solves the same decision theoretic problem with the following features. Preferences are given by β t [c t bh t ] t=0 where c t is consumption in period t and h t {0 1} is time devoted to work. If the individual begins period t not employed, then he or she will receive a job offer with probability λ w.conditionalonreceivingajoboffer,thewageassociatedwiththisofferisa random draw from the distribution with cumulative distribution function (c.d.f.) F(w). The wage associated with this job will remain fixed for the duration of the match. If the worker decides to accept the offer, he or she will begin the job in the same period as the offer was made. As in our earlier models, any match that existed in period t 1 ends with probability σ at the beginning of period t. The individual who lost a job has to wait one period before he or she starts receiving job offers with probability λ w.consumptionin each period is equal to labor earnings. Letting V(w)be the value of employment at wage w and letting U be the value of being unemployed, the Bellman equations are V(w)= w b + β[(1 σ)v (w)+ σu] and U = β [(1 λ w )U + λ w ] max(v (w) U) df(w) It is easy to show that the optimal job acceptance decision for this worker is characterized by a reservation wage, which we denote by w.onecanalsoshowthatdecreasesin

17 Quantitative Economics 1 (2010) Aggregate labor market outcomes 113 λ w or increases in σ lead to decreases in the reservation wage. The equation that governs the dynamics of the employment rate, e t,is e t+1 = (1 σ)e t + λ w (1 F(w ))(1 e t ) We focus on the steady state behavior of the unit mass of workers that each solve this problem, and in particular will ask how changes in the two frictional parameters λ w and σ affect steady state employment. For our numerical calculations, we consider a period to be a month, and set the values of β, λ w,andσ to be the same as in our calibrated model. An important consideration in comparing results across models is that one might expect the shape of the distribution to affect the results, in particular in the vicinity of the reservation wage. With this in mind, we calibrate this model so that the c.d.f. F(w) is the same as the c.d.f. for the stationary distribution of the idiosyncratic shock process from our calibrated model. We then choose the value of b so that the steady state employment rate is the same as in our calibrated model, that is, equal to These last two choices imply that the distribution of wages is the same in the two settings and that at least in an average sense, the marginal decision is in the same place in the distribution Extension 2: Adding a match termination decision The second extension is in the spirit of Mortensen and Pissarides (1994). Preferencesare the same as in Extension 1. As in Extension 1, we continue to assume that if a worker receives an offer, the wage is drawn from a distribution with c.d.f. F(w).However,ifthejob is accepted, it will evolve stochastically in future periods, according to an AR(1) process, log w t+1 = ρ log w t + ε t+1 where ε is an i.i.d. normally distributed random variable with mean zero and standard deviation σ ε. The timing is as follows. For any worker who was employed in period t 1, at the beginning of t he or she is subject to a separation that occurs with probability σ. If a separation does not occur, a new draw for ε is realized, at which point the worker decides whether to continue with the job. If not, he or she separates. A worker who experienced a separation has to wait one period before he or she starts receiving new job offers. The arrival probability of job offers is λ w.lettingg(w w) denote the c.d.f. for next period s wage given that this period s wage is equal to w, which is implied by the stochastic process for wages on the job, the Bellman equations are [ V(w)= w b + β (1 σ) ] max(v (w ) U) dg(w w) + σu 13 We solve the model using value function iteration. We use a grid with 10,000 points on w on the interval [ 2σ w 2σ w ],whereσ w is the standard deviation of the distribution described by F(w),andapplyTauchen s (1986)methodforthediscreteapproximation. 14 This cannot hold exactly, since employment decisions in our model are determined by both productivity and assets.

18 114 Krusell, Mukoyama, Rogerson, and Şahin Quantitative Economics 1 (2010) and U = β [(1 λ w )U + λ w ] max(v (w) U) df(w) The optimal search strategy for this individual will be described by a single reservation wage, which is relevant both for which new offers to accept and for which jobs to separate from. We again denote the reservation wage by w. Denote the measure of employed workers over wages by M t (w).thenthelawofmotion for the employment rate in this model is described by e t+1 = (1 σ) (1 G(w w))dm t (w) + λ w (1 F(w ))(1 e t ) The evolution of M t (w) is described by M t+1 (w ) = λ w (1 e t ) max(f(w ) F(w ) 0) + (1 σ) max(g(w w) G(w w) 0)dM t (w) We calibrate this model in a similar fashion. In particular, we set β, λ w, and σ as before. We choose the parameters ρ and σ ε that describe the stochastic evolution of wages on the job to be the same as those in our original calibration and we choose the c.d.f. F(w) to correspond to that of the stationary distribution of the stochastic process on wages. We then choose b so that the steady state employment rate is equal to Results We now consider the effects of changes in frictions on steady state employment. 16 Table 10 contains results for changes in λ w.thesecondcolumnrepeatstheresultsforour model that also appeared in Table 4. The next two columns reports results for the two extensions just described. The final column shows results for the benchmark Pissarides model used in the previous section, except that we have now calibrated λ w in this model so as to give a steady state employment rate of We begin by comparing the employment effects associated with a decrease in λ w, from the benchmark value of 428 to the value of 2.Inourmodel,thedropinsteadystate employment is 1 1,whilethecorrespondingnumbersforExtension1,Extension2,and 15 Once again, we solve the model using value function iteration. We use a grid with 240 values for w on the interval [ 2σ w 2σ w ] and apply Tauchen s (1986) method for the discrete approximation of F(w) and G(w w).thecomputationofthesteadystatee is more involved: we iterate over the descretized measures M t (w) and e t using their transition equations until they converged. 16 The effects on generalized unemployment in our model are the same as those reported earlier, so we do not repeat them. For the other models, all nonemployed workers are unemployed by standard interpretations. 17 As we move from row to row, we adjust the value of λ w for this specification proportionately. Moving from the first row to the third row, the values of λ w for the Pissarides model are 053, 038,and 018.